Modeling of Growth Kinetics
277
(7.42)
max
The definition of the cybernetic variable
in Eq. (7.41) is based on the so-called matching law
model, which specifies that the total return from allocation of resources to different alternatives is
maximized when the fractional allocation equals the fractional return. Thus in the cybernetic model
the resources are allocated for synthesis of that enzyme which gives the highest specific growth rate
(or highest Mj. Originally the cybernetic variable
w{
was also defined according to the matching law
model, but the definition in Eq. (7.42) is superior to the double matching law concept - especially
for description of simultaneous metabolism of two equally good substrates (here the double
matching concept predicts a specific growth rate that is only half of that obtained on each
substrate).
With the definition of the cybernetic models in Eqs. (7.38)-(7.42), the cybernetic model can handle
both diauxic and triauxic batch fermentations (Kompala
et al.,
1986). A major strength of the
cybernetic models is that all the parameters can be estimated on the basis of experiments on the
individual substrates, and thereafter the model does a good job in fitting experiments with mixed
substrates (see Fig. 7.15). Considering the large amount of experimental data presented by Kompala
et al.
(1986) and the quantitatively correct description of many experiments, it must be concluded
that despite their rather empirical nature cybernetic models are well suited to description of growth
on truly substitutable substrates.
According to the model of Kompala
et al.
(1986), the biomass in a chemostat that has been
subjected to feed for a long period with only one carbohydrate should contain transport enzymes
only for uptake of this particular substrate. All other enzyme systems would have been degraded or
diluted to virtually zero by the growth of the biotic phase. Thus a pulse of another carbohydrate
added to the chemostat would not be consumed, but this is contradicted by experimental
observations. To account for this weakness in the cybernetic model, Turner and Ramkrishna (1988)
introduced a term for constitutive enzyme synthesis with the rate ^ on,, in the Kompala
et al.
(1986)
model. The mass balance for the /th enzyme is then given by Eq. (7.43).
Now the microorganism is always allowed a latent capability to metabolize - at least at a low rate -
substrate different from that on which it is accustomed to grow. As a final note it can be mentioned
tiiat the cybernetic modeling concept has been extended to describe also other phenomena
prevailing during growth of microorganisms, but these extensions will not be discussed here [see
e.g. Varner and Ramkrishna (1999)]. As is the case for compartment models of Section 7.4.1 the
cybernetic models have now been developed to a stage where the complexity of the model and its
many empirical parameters obscures the main object of the exercise, namely to provide a
reasonable verbal model to explain certain observations. The simple, semi-mechanistic model
(7.43)
previous page 300 Bioreaction Engineering Principles, Second Edition  read online next page 302 Bioreaction Engineering Principles, Second Edition  read online Home Toggle text on/off